Big Bang- Big Crunch Optimization in Second Order Sliding Mode Control

21 International Journal for Modern Trends in Science and Technology
Volume: 2 | Issue: 03 | March 2016 | ISSN: 2455-3778IJMTST
Big Bang- Big Crunch Optimization in
Second Order Sliding Mode Control
TejasKulkarni
Department of Electrical Engineering, G.H.Raisoni Institute of Engineering and Technology, Pune,
India.
Paper Setup must be in A4 size with Margin: Top 1.1 inch, Bottom 1 inch, Left 0.5 inch, Right 0.5 inch,
In this article, Second order sliding mode with Big Bang- Big Crunch optimization technique is employed
for nonlinear uncertain system.The sliding surface describes the transient behavior of a system in sliding
mode. Frequently, PD- type sliding surface is chosen as a hyperplane in the system state space.An integral
term incorporated in the sliding surface expression that resulted in a type of PID sliding surface as hyperbolic
function for alleviating chattering effect. The sliding mode control law is derived using direct Lyapunov
stability approach and asymptotic stability is proved theoretically. Here, novel tuning scheme is introduced for
estimation of PID sliding surface coefficients, due to which it reduces the reaching time as well as disturbance
effect.The simulation results are presented to make a quantitative comparison with the traditional sliding
mode control. It is demonstrated that the proposed control law improves the tracking performance of system
dynamic model in case of external disturbances and parametric uncertainties.
KEYWORDS: Second order sliding mode control, Big Bang- Big Crunch (BBBC) optimization, PD sliding
surface, PID and Lyapunov stability.
Copyright © 2015 International Journal for Modern Trends in Science and Technology
All rights reserved.
I. INTRODUCTION
Sliding Mode Control (SMC) is known to be a
robust control method appropriate for controlling
uncertain systems. High robustness is maintained
against various kinds of uncertainties such as
external disturbances and measurement error.
The dynamic performance of the system under the
SMC method can be shaped according to the
system specification by an appropriate choice of
switching function. Robustness is the most
excellentbenefit of a SMC and systematic design
procedures are well known in available literature
[1-3].Even though, SMC is a powerful control
method that can produce a robust closed-loop
system under plant uncertaintiesand external
disturbances, it suffers from lots of shortcomings.
In idealSMC, the switching of control occurs at
infinitely highfrequency to force the trajectories of
a dynamic system to slide along therestricted
sliding mode subspace. In practice, it is not
possible to change the control infinitely fast
because of the time delay for control computations
andphysical limitations of switching devices. As a
result, the SMC action can lead to high frequency
oscillations called chattering whichmay excite
unmodeled dynamics, energy loss, and system
instability andsometimes it may lead to plant
damage.Creation of a boundary layeraround the
sliding manifold and the use of observers were
attempted for chattering removal.However, the
control still acts on the derivative of the sliding
variable. Emel'yanov et al.[4] initially presented
the idea of acting on the higher derivatives of the
sliding variable andprovided second order sliding
algorithms such as the twisting algorithm, and
algorithm witha prescribed law of convergence [5].
Another so called real second order drift
algorithmwas presented in Emel'yanov et al. [6].
The term real means that the algorithm has a
meaningonly when there is a switching delay. In
the ideal case of infinite frequency switching,
ABSTRACT

Big Bang- Big Crunch Optimization in Second Order Sliding Mode Control
itceases to converge. The so calledsuper twisting
algorithm, which is applicable to systemswith
relative degree one with respect to the sliding
variable, [7] completely removes chattering.Levant
[8] reported that in second order sliding, the
sliding accuracy is proportionalto the square of
the switching time delay which turns out to be
another advantage of higherorder sliding
modes.Since then, a number of such controllers
have been developed in the literature and are
currently finding useful applications.Bartolini et
al. presented a sub-optimal second ordersliding
algorithm in [9-11].
The higher order sliding mode concept was
applied to thecontrol of constrained manipulators
in Bartolini et al. [12]. Levant et al. [13] have
applied itto aircraft pitch control and the same
idea has been implemented for robust exact
differentiationin [14]. Levant [15] demonstrated
that the twisting algorithm with a variable
integrationstep size performs better in the
presence of measurement errors.Eker [16]
proposed second order sliding mode control based
on a PID sliding surface with independent gain
coefficients and address issues related to SMC
implementation on practical electromechanical
plant. The second-order sliding mode
correspondsto the control acting on the second
derivative of the sliding surface[16]. It should be
noted that different 2-SMC algorithmshave been
presented in the literature such as the `twisting'
and`super-twisting' [17], `sub-optimal' and `global'
[18,19], asymptoticobserver algorithms [20], and
drift algorithm [21]. Readers arereferred for the
extensive information about these methods to[18,
21,22].The outline of this paper can be
summarized as follows: Section 2 presented
traditional sliding mode control for nonlinear
system. In section 3, the design of second order
sliding mode with PID sliding surface based on Big
Bang- Big Crunch optimizationintroduced. Then,
Section 4 describes the close loop stability
analysis based on Lyapunov theory.
MATLAB/Simulink based numerical simulations
for stabilizing second order nonlinear system
presented in Section 5. Finally, conclusions are
given in Section 6.
II. FIRST ORDER SLIDING MODE CONTROL
Consider a second order nonlinear system,
which can be represented by the following
equation
( ) ( ) ( )x f x g x u d t   (1)
where,  
T n
x x x R  is the state vector with
initial condition 0 0( )x x t , ( )f x and ( )g x are the
nonlinear dynamic function, m
u R is the control
input vector and ( )d t is the external disturbance.
The control problem is to synthesize a control law
u such that state vector x traces the desired
trajectory  
T
d d dx x x  within the tolerance error
bound defined by
1 2 1 2, , 0, 0d dx x x x          (2)
It is assumed that, ( )dx t , ( )dx t and ( )dx t are well
defined and bounded for all time t . Tracking error
is defined as de x x  and Here, sliding mode is
said to be first order sliding mode if and only if S(t)
= 0 and ( ) ( ) 0S t S t  is the fundamental sliding
condition for sliding mode. The main objective of
first order SMC is to force the state to move on the
switching surface S(t) = 0. Let us consider the
sliding surface ( )S t be expressed as in the state
space
2
R by ( ; ) 0S x t 
1
( )
n
d
S e t
dt


 
  
 
(3)
Where, n denotes order of uncontrolled system
and λ is positive constant. The nonlinear system is
having second order ie n=2 and then, in a two
dimensional phase plane related to the sliding
surface equation, the solution is a straight line
passing through the origin, afterwards
maintaining the error on sliding surface will result
in e(t) approaching zero. Therefore, S(t) =0 is a
stable sliding surface and e(t)0 and t∞. The
derivative of sliding surface is given as follows,
( ) ( )t tS e e    (4)
The second time derivative of error can be written
in terms of the plant parameters
( ) ( ) ( )t d de x x f x g x u x        (5)
The above equation substituted in Eq. (4)
( ) ( ) ( )t dS e f x g x u x      (6)
The control input can be given as
( ) ( ) ( )eq swu t u t u t  (7)
Where, ( )equ t is equivalent control and ( )swu t is
switching control, respectively. The equivalent
control is based on system parameters with
disturbance assumed to zero. While switching

control ensures the discontinuity of the control
law across sliding surface, providing additional
control to account for the presence of matched
disturbances and unmodeled dynamics.
The equivalent control is derived from sliding
condition, When Eq.(6) is become zero, which is
the necessary condition for the tracking error to
remain on the sliding surface.
 ( )
1
( ) ( )
( )
eq t du t e f x x
g x


    (8)
The switching input is introduced as follows
( ) sgn( ( ))sw du t k S t (9)
Where, dk is a positive constant and sign(.)
denotes sign function defined as follows
1 0
s ( ) 0 0
1 0
s
ign s s
s


 
 
(10)
The signum function in the hitting control Eq.(9)
requires infinite switching on the part of the
control signal and actuator to maintain the system
dynamics on the sliding surface. In practice,
actuator limitations, transport delays,
computational delays and other factors prevent
true sliding leads to phenomenon known as
chattering effect in which the states jump back
and forth across the sliding manifold while
decaying towards the origin. This effect can be
eliminated by using second order sliding mode
control technique.
III. SECOND ORDER SLIDING MODE CONTROL
OPTIMIZATION BY USING BBBC
In the higher order sliding mode controller
category, the second order sliding mode controller
(SOSMC) has been extensively developed. This is
compared to the first order sliding mode control
has advantage of lesser chattering with better
convergence accuracy. The main idea of SOSMC is
not only to force the sliding surface but also its
second order derivatives to zero.
Let us consider PID sliding surface with constant
coefficients expressed as
0
( ) ( ) ( ) ( ) ( )
t
P I DS t S t K e t K e t dt K e t     (11)
Where, ( )e t is the tracking error, ( )e t is the time
derivative of ( )e t , dx the desired trajectory.
,P IK K and DK are designed such that the sliding
mode on 0S  is stable ie the convergence of S to
zero in turn guarantees that x and x also
converge to zero. The coefficients of PID based
sliding surface are strictly positive constant
,P IK K and T
DK R . Taking the derivative of
sliding surface with respective time, defined by Eq.
(11) can be given as
( ) ( ) ( ) ( ) ( )P I DS t S t K e t K e t K e t       (12)
A necessary condition for the output trajectory to
remain on the sliding surface ( )S t is ( ) 0S t 
( ) ( ) ( ) 0P I DK e t K e t K e t    (13)
The control effort being derived as the solution of
( ) 0S t  without considering lumped uncertainty
( ) 0L t  is to achieve the desired performances
under the nominal model and it is referred to as
equivalent effort represented by equ
 1
( ) ( ( )) ( ( ) ( ) ) ( ) ( )eq D D d P Iu t K g x K f x d t x K e t K e t
       (14)
If unpredictable perturbations from the parameter
variations or external load disturbance occur, the
equivalent control effort can’t ensure the favorable
control performance. Thus the auxiliary control
effort should be designed to eliminate the effect of
unpredictable perturbations. The auxiliary control
effort is referred to as hitting control effort
represented by disu .
In SOSMC, disu is given as follows
( ) sgn( ( ))dis Du S t K S t   (15)
Where, DK is a hitting control gain concerned
with the upper bound of uncertainties and
sgn(.) is a sign function.The controller must be
designed such that it can drive the output
trajectory to the sliding mode ( ) 0S t  . The output
trajectory, under the condition that it will move
towards and reach the sliding surface, is said to be
on reaching phase.
Estimation of hitting gain and sliding surface
coefficients:
In the proposed controller the sliding surface
differential is used and DK is calculated with an
adaptive equation. Using the proposed controller
the results caused more adequate effect on the
smoothness of the control input, compared with
sliding mode control in the initial time.It is

assumethe modelingerroris boundedandtheupper
boundis 0 1 ( )d d e t , where d0, d1 are positive
constant and disturbance isbounded d  .
Defining a new parameter  as given
0 1 ( )i d d e t    (16)
The discontinuous gain can be expressed as
ˆ
D i iK    (17)
Where,
i is positive constant and adaptive
control is given by [23]
ii ef  ~ˆ 

(18)
Here, [ , ]s s    and fis positive constant used for
adaptation of discontinuous gain in SOSMC.
In this control algorithm sliding surface
coefficients ,P IK K and DK are are determined by
using BBBC optimization algorithm.Here
proportional gain is assumed to be unity, But
integral gain and derivative gain are estimated.
The BBBC method was developed and proposed as
a novel optimization method by Erol and Eksin
(2006) This method is derived from one of the
evolution of the universe theories in physics and
astronomy, describing how the universe was
created, evolved and would end, namely the BBBC
phase. The big bang phase comprising random
energy dissipation over the entire search space or
the transformation from an ordered state to a
disordered or chaotic state. After the big bang
phase, a contraction occurs in the big crunch
phase. Here, the particles that are randomly
distributed are drawn into an order. This aims to
reduce the computational time and have quick
convergence even in long, narrow parabolic
shaped flat valleys or in the existence of several
local minima.
The big bang phase is somewhat similar to
creation of initial random population in GA. The
designer should handle the impermissible
candidates at this phase. Once the population is
created, fitness values of the individuals are
calculated (Genç, Eksin, & Osman, 2010). The
crunching phase is a convergence operator that
has many inputs but only one output, which can
be referred to as the center of “mass.” The center of
mass represents the weighted average of the
candidate solution positions. Here, the term mass
refers to the inverse of the merit function value
(Singh &Verma, 2011). After a number of
sequential banging and crunching phases the
algorithm converges to a solution. The point
representing the center of mass “Xc” of the
population is calculated according to the formula
1
0
1
0
1
N
k
k
C
N
k
k
Xk
f
X
f







(19)
whereXk is a point within an n-dimensional search
space generated, here it is related to the
numerator polynomial coefficients, fk is a fitness
function or objective value of the candidate k and
N is the population size in banging phase. The
convergence operator in the crunching phase is
different from wild selection since the output term
may contain additional information (new
candidate or member having different parameters
than others) than the participating ones. In the
next cycle of the big bang phase, new solutions are
created by using the previous knowledge (center of
mass), the fitness function fk (Erol&Eksin, 2006) is
21
0
[ ( ) ( )] ;
M
k r
i
T
f y i t y i t M
t


    

 (20)
Where, y(iΔ t) and yr(iΔ t) are the unit step
responses of the higher-order and the
reduced-order models at time t=Δ t. Usually time T
is taken as 10 s and Δ t=0.1 s.
The basic BBBC algorithm utilized here is as
follows:
1. The Big bang starts by generating the new
population. [Start
2. For each iteration the algorithm will act
such [Evaluate Fitness value that each
candidates will move in a direction to
improve its fitness function. The action
involves movement updating of individuals
and evaluating the fitness function for the
new position.
3. Compare the fitness function of the new
[Compare Fitness Function position with
the specified fitness function. Repeat the
above steps for the whole set of candidates.
4. Stop and return the best solution if
maximum [Maximum iteration iteration
is reached or a specified termination

criterion is met. Else, update and start
generating new population at step 1.
5. Go to step 2 for fitness evaluation. [Loop
IV. NUMERICAL SIMULATION AND EXPERIMENTAL
RESULTS
Speed control of DC Motor is chosen as an
example to demonstrate the effectiveness of the
proposed control algorithm as SOSMC with BBBC
optimization. A simple model of a DC motor driving
an inertial load shows the angular rate of the load,
ω(t), as the output and applied voltage, Vapp, as the
input. The ultimate goal of this example is to
control the angular rate by varying the applied
voltage. Fig. 1 shows a simple model of the DC
motor driving an inertial load J.In this model, the
dynamics of the motor itself are idealized; for
instance, the magnetic field is assumed to be
constant. The resistance of the circuit is denoted
by R and the self-inductance of the armature by L.
The important thing here is that with this simple
model and basic laws of physics, it is possible to
develop differential equations that describe the
behavior of this electromechanical system. In this
example, the relationships between electric
potential and mechanical force are Faraday's law
of induction and Ampere’s law for the force on a
conductor moving through a magnetic field.A set
of two differential equations given in (21) describe
the behavior of the motor. The first for the induced
current, and the second for the angular rate,
Fig. 1 DC motor driving inertial load
1
( ) ( )b
app
Kdi R
i t t V
dt L L L
      
( ) ( )mF KKd
t i t
dt J J

    (23)
Above state equation can be written in a second
order uncertain form
Where, is the output, , u(t) is the
control input and are the nominal plant
parameters, are the unknown model
uncertainties.
(25)
Where denotes the lumped
uncertainties.
In this simulation the nominal parameter values
are calculated as
with step input signal (armature voltage) with an
amplitude of 4.5 V and output is the speed of DC
motor in rpm.As observed in Fig. 2 even in the
presence of external disturbances, the proposed
control strategy is able to provide better trajectory
response compared with conventional SMC. For
comparative analysis, the phase portrait obtained
with SOSMC based on PID sliding surface is
presented in Fig. 3. Chattering problem was
eliminated in SOSMC method, because of
hyperbolic function of sliding surface and
differentiator. The improved performance of
SOSMC with BBBC optimization for sliding
hyperplane over the traditional SMC as shown in
Fig. 4, which is due to the ability of recognize and
compensate the external disturbances with BBBC
tuning approach to determine sliding coefficients.
Despite the external disturbance forces and
uncertainties with respect to model parameters,
the proposed control technique allows the DC
motor to track the desired speed trajectory with a
small tracking error as presented in Fig.5. In the
above simulation, the parameters of the
controllers were chosen based on the assumption
that exact values of plant parameters are not
known, but with a maximal uncertainty of ± 10%
over the previous adopted values. For the dynamic
model simulation of electromechanical plant,
parameter values were selected as mentioned
above. From the observation of simulated results,
it was clear that SOSMC based PID sliding surface
with BBC tuning approach was employed for
avoiding oscillation in the output response when
load disturbance force was applied and also
reduce the reaching time.

Fig. 3 Phase trajectory response of DC Motor
Fig.4 Set point tracking response of DC motor
Fig.5 sliding surface response of DC motor
V. CONCLUSION
In this paper, SOSMC with PID sliding
surface was employed to deal with the speed
regulation of DC motor. An integral term
expressed in the PD-type sliding surface that
resulted in a type of PID sliding surface. Due to the
integral term chattering problem was not appear
in the phase trajectory response. In proposed
control algorithm, hitting gain is obtained from
adaptive law along with BBBC optimization
technique utilized to determine PID sliding
surface coefficients.
The disturbance effect was identified and
compensated by using optimized SOSMC
technique. The stability and convergence
properties of closed loop systems were analyzed
through phase trajectory response. Through
numerical simulations, the improved performance
of proposed control technique over the
conventional SMC was demonstrated.
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